An approximation scheme for the effective Hamiltonian and applications

Authors:
M. Rorro
Affiliations:
Dipartimento di Metodi e Modelli Matematici Università di Roma "La Sapienza", Roma, Italy
Venue:
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
Year:
2006

Citing 5
Cited 1

On the large time behavior of solutions of Hamilton—Jacobi equations

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Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods

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Computing the Effective Hamiltonian in the Majda--Souganidis Model of Turbulent Premixed Flames

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Convergence of Semi-Lagrangian Approximations to Convex Hamilton-Jacobi Equations under (Very) Large Courant Numbers

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Computing the Effective Hamiltonian Using a Variational Approach

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A numerical study for the homogenisation of one-dimensional models describing the motion of dislocations

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Abstract

We introduce an approximation for first order Hamilton-Jacobi equations with a convex Hamiltonian periodic in the space variable. To this end we use a first order semi-Lagrangian scheme to compute a solution of the so-called cell problem and the related effective Hamiltonian. The scheme can also be interpreted as a discrete version of the Lax-Oleinik representation formula for the exact viscosity solution of the time-dependent problem. The information included in the solutions of the cell problem and in the effective Hamiltonian is exploited for the approximation of the Aubry set. We prove some properties of the scheme and illustrate the effectiveness of our approximation by several tests in dimension 1 and 2.